Finite Group Databasebeta

← C2×#27703 #6457064

Subgroups — C2×#27703 #6457064

Order 512.

2124 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2439C211yesmaximal, characteristic, complemented
3#27703C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2101C211yesmaximal, characteristic, complemented
6C2×#2073C211yesmaximal, characteristic, complemented
7C2×#2023C211yesmaximal, characteristic, complemented
8C2×#1993C211yesmaximal, characteristic, complemented
9C2×#1936C211yesmaximal, characteristic, complemented
10C2×#1934C211yesmaximal, characteristic, complemented
11C2×#1895C211yesmaximal, characteristic
12C2×#1875C211yesmaximal, characteristic
13C2×#1835C211yesmaximal, characteristic, complemented
14C2×#1823C211yesmaximal, characteristic, complemented
15C2×#1753C211yesmaximal, characteristic, complemented
16C2×#1736C211yesmaximal, characteristic, complemented
17C23×M16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2504—24no—
30#2504C2218yes—
31#2439—24no—
32#2439C2218yescomplemented
33C2×#397—21no—
34C2×#397—21no—
35C2×#396—21no—
36C2×#393—21no—
37C2×#391—21no—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#387—21no—
43C2×#386—21no—
44C2×#386—21no—
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2218yescomplemented
51#2101C2218yescomplemented
52#2073C2218yescomplemented
53#2023C2218yescomplemented
54#1993C2218yescomplemented
55#1936C2218yescomplemented
56#1934C2218yescomplemented
57#1895C2218yes—
58#1875C2218yes—
59#1835C2218yescomplemented
60#1823C2218yescomplemented
61#1753C2218yescomplemented
62#1736C2218yescomplemented
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C22×M16—24no—
83C22×M16—24no—
84C22×M16—42no—
85C22×M16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8C2314yes—
94C2×#121—22no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121C2314yes—
98C2×#117—22no—
99C2×#117—24no—
100C2×#117—42no—
101C2×#117C2314yes—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—42no—
132C2×#102—42no—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8—24no—
138C2×C22⋊C8C2314yes—
139C2×#97—42no—
140C2×#97C2314yes—
141C2×C4×C8—24no—
142C2×C4×C8C2314yes—
143#397—28no—
144#397—28no—
145#396—28no—
146#393—28no—
147#391—28no—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#387—28no—
157#386—28no—
158#386—28no—
159#382—22no—
160#382—22no—
161#382—22no—
162#382C2314yes—
163#382C2314yes—
164#382C2314yes—
165#380—22no—
166#380C2314yes—
167C4×C22⋊C4D414yes—
168C4×C22⋊C4C2314yes—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95D411yescharacteristic
179C2×#95D411yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C25C2×D411yescharacteristic
196C22×Q8—24no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—82no—
200C22×D4—24no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—82no—
226C23×C4—82no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4#4114yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C2×M16—28no—
238C2×M16—28no—
239C2×M16—44no—
240C2×M16—416no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8#4114yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260#121—28no—
261#121—28no—
262#121—48no—
263#117—28no—
264#117—44no—
265#117—48no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4#4114yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D414yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333#102—28no—
334#102—28no—
335#102—44no—
336#102—44no—
337C22⋊C8—28no—
338#97—44no—
339C4×C8—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95C2×D414yes—
350#95C2×D414yes—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24#12414yes—
366C24C22×D412yes—
367C24C22×D411yescharacteristic
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—84no—
371C2×Q8—816no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—816no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—816no—
445C22×C4#12414yes—
446C22×C4#12418yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462M16—416no—
463C2×C8—28no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8C2×#4114yes—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—84no—
503C4⋊C4C2×#4114yes—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—48no—
520C23—28no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23C2×#12414yes—
548C23C2×#12412yes—
549C23#54612yes—
550C23#54712yes—
551C23#57512yes—
552C23#57512yes—
553C23#57612yes—
554C23#57712yes—
555C23C23×D411yescharacteristic
556Q8—816no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—816no—
562D4—84no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—216no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—816no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4C2×#12414yes—
617C2×C4C2×#12412yes—
618C2×C4#52914yes—
619C2×C4#53914yes—
620C2×C4#54612yes—
621C2×C4#54712yes—
622C2×C4#55114yes—
623C2×C4#55214yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C8—24no—
631C8—42no—
632C8—42no—
633C8—42no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—416no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—168no—
650C22#248814yes—
651C22#313614yes—
652C22#314014yes—
653C22#332214yes—
654C22#332614yes—
655C22#334714yes—
656C22#335014yes—
657C22C22×#12411yescharacteristic
658C22C2×#54611yescharacteristic
659C22C2×#54711yescharacteristic
660C22C2×#57511yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57611yescharacteristic
663C22C2×#57711yescharacteristic
664C4—24no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#248814yes—
677C4C2×#52911yescharacteristic
678C4C2×#53911yescharacteristic
679C4C2×#55111yescharacteristic
680C4C2×#55211yescharacteristic
681C2—44no—
682C2—162no—
683C2#2770318yesdirect product factor
684C2C2×#248811yescharacteristic
685C2C2×#313611yescharacteristic
686C2C2×#314011yescharacteristic
687C2C2×#332211yescharacteristic
688C2C2×#332611yescharacteristic
689C2C2×#334711yescharacteristic
690C2C2×#335011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2439C2semidirect
3. #27703C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2101C2semidirect
6. C2×#2073C2semidirect
7. C2×#2023C2semidirect
8. C2×#1993C2semidirect
9. C2×#1936C2semidirect
10. C2×#1934C2semidirect
13. C2×#1835C2semidirect
14. C2×#1823C2semidirect
15. C2×#1753C2semidirect
16. C2×#1736C2semidirect
32. #2439C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2073C22semidirect
53. #2023C22semidirect
54. #1993C22semidirect
55. #1936C22semidirect
56. #1934C22semidirect
59. #1835C22semidirect
60. #1823C22semidirect
61. #1753C22semidirect
62. #1736C22semidirect
683. C2#27703direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25042
C2×#24392
#277032
C2×#21012
C2×#20732
C2×#20232
C2×#19932
C2×#19362
C2×#19342
C2×#18952
C2×#18752
C2×#18352
C2×#18232
C2×#17532
C2×#17362